Issue with lighting and rendered environment map
I believe your issue is that you are using the Blender Render engine instead of Cycles (much newer). This would definitely lead to the darker bottom faces because Blender Render does not utilize bounce lighting (in real life, if you have a desk with a cavity under it and a white carpet floor for example, when photons hit the carpet, they bounce and also hit the underside of the desk, illuminating it). Blender Render also cannot do proper environment maps.

To fix all of these problems, go to the top bar and where it says Blender Render, instead select Cycles Render from the dropdown:

 --> 

Select the camera and go to the Camera Object Data tab in the Properties panel. Change the settings under Lens to Panoramic and Equirectangular. This will create a true world image.



Go to the Render Settings tab in the Properties panel. Under Light Paths, change the settings to these. These are the optimal bounce lighting settings that I use in almost every render. While on the subject of lighting, I would highly recommend repositioning your lamps and changing their strength and color so that the image looks better.



Go to Properties panel > Render Settings > Sampling and change the Render: samples to 250. Also change the Clamp Indirect: value to 3. Both of these will improve your image's clarity.

Go to Properties panel > Render Settings > Dimensions and change the X: and Y: dimensions to 2000x1000. Up the percentage slider to 100%.



Lastly, since I assume you want a panorama like this:



...and not like this:



...change your camera positioning in the 3D View Properties panel to this:



After adjusting the lighting as well, I was able to make this:



Also, here is the adjusted .blend file: